Projection of a Sphere

Stereographic projection from a point on a sphere maps the sphere to the
plane so that circles on the sphere are projected to circles in the plane,
with circles that pass through the point of projection mapping to straight
lines in the plane. Here we see colored circular bands on a sphere
projected to the plane below; one of the bands (the one containing the point
of projection) extends out to infinity. This example is the first step to
understanding the projections of the torus that appears in In- and Outside the Torus and Hopf-Link Torus.

The movie The Hypersphere: Projections and Foliations, from which
this image is taken, describes these ideas more fully.